Electronic Transactions on Numerical Analysis
its book of mathematics on numerical analysis.
15 Counting
Let us examine a reallife problem. As a university professor I feel that teaching twenty
students for one hour a week is too much of a demand on my time and that it cuts into my
research (on sour mash). Therefore, in order to cut back on my teaching load in future classes,
I have decided to flunk most of my students this semester. Specifically, I am going to give one
A, one B, one C, one D, and I'll flunk everyone else.
16 Multinomials
Let us slightly generalize the problem of selecting r out of n objects without respect to
order. Suppose, for example, that I have 20 students.
17 Stirling Numbers of the Second Type
Again: a partition of n objects is a division of these objects into separate classes. Each
object must be in one and only one class and partitions with empty classes are not allowed.
18 Matrices
Matrix algebra is also known as linear algebra and it is one of the most important
disciplines of mathematics. It pervades virtually all of mathematics, but it could be argued to
be closest to geometry. In this chapter, we will look at the relation between matrices and graphs.
19 Boolean Algebra
It has been known for sometime in mathematics that set algebras and Boolean1 algebras
are different perspectives on the same thing. The treatment of sets here is informal and is
known as naive set theory. Most of the time naive set theory is sufficient for the purposes of
even professional mathematicians.
20 Probability and the Law of Addition
We are interested in the probabilities of events. Whatever events are, they are not
numbers. Therefore, given events A and B, if I speak of the event A + B, it does not mean A
plus B. Rather it is shorthand for A or B. Elsewhere,
21 PIE
The law of addition of Section 20 is a probabilistic interpretation of the principal of
inclusion and exclusion. In general, the problem of counting things is to count each object once
and only once. Suppose we have N objects, some of which have property a and some which
have property b and some of which have both properties.
23 The Law of Multiplication
Probability trees are one of the most useful
tools in probability and applied mathematics. We
will use them throughout the rest of this text. The
graph in Figure 1 is a probability tree.
24 The Bonferroni Inequality
1By far its most useful application is in joint confidence intervals. The inequality gives
you a confidence interval without assuming independence of the various parameters. It usually
turns out at around 95% confidence that the confidence region isn’t much smaller than with the
assumption of independence.
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